This is the first topic of CTET Mathematics Preparation. These slides cover Number system, Multiplication, division, Algebra formula and LCM & HCF.

1. Mathematics
Introduction
&
Formulas

2. Number System
• Number System is a writing system for
expressing numbers.

3. • Real Number:- When both rational and irrational
numbers combined, the combination is defined as
real number. Real numbers can be positive and
negative numbers.

4. Classification of Real Number
• Natural Numbers:- includes all the counting
numbers like 1,2,3,4,……..
• Whole Numbers:- all natural numbers including 0.
• Integers:- all natural numbers and negative of
natural numbers including 0. e.g., -2,-1,0,1,2,…
(note:- 0 is neither positive nor negative.)
• Rational Numbers:- numbers that can be written in
the fraction form, p/q, where q can’t be 0.
• Irrational Numbers:- an irrational number can’t be
expressed in the form of p/q, where q can’t be 0.

5. Some Important Formulae
• Sum of first n natural numbers.
1+2+3+………+n = 1/2n(n+1)
• Sum of square of first n natural numbers.
12+22+32+……+n2 = 1/6n(n+1)(2n+1)
• Sum of cubes of first n natural numbers.
13+23+33+…..+n3 = {1/2n(n+1)}2
• Sum of first n odd numbers = n2
• Sum of first even numbers = n(n+1)
• Sum of odd numbers from 1 to n = (n+1/2)2
• Sum of even numbers from 1 to n = n/2(n/2+1)

6. Multiplication
(sign concept)
Multiplication of integers is similar to multiplication
of whole numbers except the sign of the product
needs to be determined.
If both numbers are positive, the product will be
positive.
If both numbers are negative, the product will be
positive.
If any number is negative, the product will be negative.
In other words, if the sign are same the product will be
positive, if they are different the product will be
negative

7. Division based formulae
• Dividend = divisor * quotient + remainder
Divisor = is the number to be divided with, also said
denominator .
Dividend = is the number to be divided, also said
numerator.
Quotient = is the result found after division.
• Example:- a divided by b, a/b
here, a is dividend or numerator
b is divisor or denominator
• Divisor = dividend - remainder/ quotient.
• Quotient = dividend - remainder/ divisor

8. Algebraic Formulae
• (a + b)2 = a2 + 2ab + b2
• (a - b)2 = a2 - 2ab + b2
• a2 - b2 = (a + b)(a - b)
• (a + b)3 = a3 + b3 + 3ab(a + b)
• (a - b)3 = a3 - b3 - 3ab(a - b)
• a3 + b3 = (a + b)(a2- ab + b2)
• a3 - b3 = (a - b)(a2 + ab + b2)
• (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
• a3 + b3 + c3 = (a + b + c)(a2 + b2 + c2 – ab – bc - ca)+3abc
• (a + b)2 + (a - b)2 = 2(a2 - b2)
• (a + b)2 – (a - b)2 = 4ab

9. • If a + b + c = 0, then a3 + b3 + c3 = 3abc
• (x + a)(x + b) = x2 + (a + b)x + ab
• (x + a)(x - b) = x2 + (a - b)x – ab
Laws of Exponent
• (am)(an) = am+n
• (ab)m = ambm
• (am)n = amn
• a0 = 1
• am/an = am-n
• √a = a1/2
• n√a = a1/n
• a-n = 1/an

10. HCF & LCM
• HCF (Highest Common Factor):- the HCF of two or more
numbers is the greatest number that divides each of them
exactly.
The common factor of 12 and 18 is 1,2,3,6.
The largest common factor is 6, so this is the HCF of 12 and 18.
• LCM (Lowest Common Multiple):- the LCM is the smallest
number that is a common multiple of two or more numbers.
• Product of two number = Product of their HCF & LCM.
• HCF and LCM of Fractions:
• HCF of Fractions = HCF of numerator ÷ LCM of denominator
• LCM of Fractions = LCM of numerator ÷ HCF of denominator